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Curvature of Universal Bundles of Banach Algebras

  • Maurice J. Dupré
  • James F. Glazebrook
  • Emma Previato
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

Given a Banach algebra we construct a principal bundle with connection over the similarity class of projections in the algebra and compute the curvature of the connection. The associated vector bundle and the connection are a universal bundle with attendant connection. When the algebra is the linear operators over a Hilbert module, we establish an analytic diffeomorphism between the similarity class and the space of polarizations of the Hilbert module. Likewise, the geometry of the universal bundle over the latter is studied. Instrumental is an explicit description of the transition maps in each case which leads to the construction of certain functions. These functions are in a sense pre-determinants for the universal bundles in question.

Keywords

Hilbert module Banach Grassmannian similarity class polarization universal bundle connection curvature 

Mathematics Subject Classification (2000)

Primary 46M20 37K20 Secondary 58B99 58B25 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Maurice J. Dupré
    • 1
  • James F. Glazebrook
    • 2
    • 3
  • Emma Previato
    • 4
  1. 1.Department of MathematicsTulane UniversityNew OrleansUSA
  2. 2.(Primary Inst.) Department of Mathematics and Computer ScienceEastern Illinois UniversityCharlestonUSA
  3. 3.(Adjunct Faculty) Department of MathematicsUniversity of Illinois at UrbanaUrbanaUSA
  4. 4.Department of Mathematics and StatisticsBoston UniversityBostonUSA

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